LMIs in Control/pages/L2 gain of Lure systems

The System

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q&=C_{q}x,\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}

The matrices $A,B_{p},B_{w},C_{q},C_{z}$ .

The following semi-definite problem should be solved.

{\begin{aligned}&\min _{\{P\succ 0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0\}}\gamma ^{2}\;\\&\quad \quad \quad \quad \quad \quad \quad \quad s.t.\quad {\begin{bmatrix}A^{\top }P+PA+C_{z}^{\top }C_{z}&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T&PB_{w}\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T&\Lambda C_{q}B_{w}\\B_{w}^{\top }P&B_{w}^{\top }C_{q}^{\top }\Lambda &-\gamma ^{2}I\end{bmatrix}}\preceq 0\\\end{aligned}}

The value function returns the square of the smallest provable upper bound on the ${\mathcal {L}}_{2}$ gain of the Lure's system.

The Lyapunov function which is used to proof is similar to the one for the systems with unknown parameters.

LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.